Wilfried Bony snatched a dramatic last-minute winner as Swansea City heaped fresh misery on Manchester United with a shock 2-1 success in the FA Cup third round on Monday (AEDT).

11 points off the pace in the Premier League, where they have already lost six games, United crashed out at the first hurdle in the FA Cup to give beleaguered manager David Moyes even more problems.

To compound matters, United's injury list swelled further after Rio Ferdinand had to go off with a knee injury, while Fabio da Silva was sent off shortly after coming on to replace him.

"We were a bit unlucky to lose, but we probably didn't do enough to win it in the end," said Moyes, whose side visits Sunderland in the League Cup semi-finals on Wednesday.

"We made some opportunities, (but) not enough. When we went down to 10 men, we just needed to see it out and at least take a replay if that was what had to come about, but we didn't quite get that far."

In a game preceded by a minute's applause for late Portugal great Eusebio, who died on Sunday, Swansea took a 12th-minute lead through Wayne Routledge, who cantered on to a pass from Alejandro Pozuelo to lob Anders Lindegaard.

The hosts equalised four minutes later, Javier Hernandez volleying in a cross from Alexander Buttner, but they could not find a second goal.

Ferdinand had to hobble off due to injury late on and his replacement, Fabio, was sent off for a high challenge on Jose Canas shortly after coming on, which Moyes said was the right call by the referee.

Worse was to follow at the death, with Routledge lifting a cross into the box from the left and Bony powering home a near-post header to add a new chapter of disappointment to the tale of United's unravelling season.

When we went down to 10 men, we just needed to see it out and at least take a replay ... but we didn't quite get that far.

David Moyes

West Ham United also fell, crashing to an embarrassing 5-0 defeat at second-tier Nottingham Forest, but Chelsea and Liverpool both progressed against lower-league opponents.

Chelsea beat in-form Championship side Derby County 2-0, with Liverpool avenging its shock defeat by third-tier Oldham Athletic in last season's competition with a 2-0 success at Anfield.

Chelsea had to wait until the 66th minute before making the breakthrough against former England manager Steve McClaren's side.

John Mikel Obi glanced home a header from a Willian cross to mark his 300th Chelsea appearance with his fourth goal for the club before Oscar got on the score-sheet soon after.

Chelsea manager Jose Mourinho highlighted the importance of Mikel's goal, saying: "It was important for us because we were dominating and dominating, creating and creating, but the goal was not arriving. It was like the winning goal."

Liverpool will visit either Bournemouth or Burton Albion in round four after seeing off Oldham, who stunned Brendan Rodgers's side 3-2 at Boundary Park in the fourth round last season.

Iago Aspas, a close-season signing from Celta Vigo, broke the deadlock in the 54th minute to claim his first goal for the club but Liverpool did not secure victory until Oldham's James Tarkowski put through his own goal with eight minutes to play.

West Ham became the second Premier League team to be eliminated by lower-league opposition in this season's competition when being crushed 5-0 by Championship side Forest at the City Ground.

A day after Aston Villa fell to third-tier Sheffield United, West Ham were undone by a 14-minute hat-trick from Forest forward Jamie Paterson, sandwiched by an early Djamel Abdoun penalty and a late Andy Reid strike.

Sunderland overcame third-tier Carlisle United 2-1 at the Stadium of Light to take their place in round four.

Australians impress in FA Cup wins

Australian goalkeepers Mark Schwarzer and Brad Jones made the most of rare starts, keeping clean sheets to help guide their clubs into the fourth round.

Schwarzer, 41, impressed in Chelsea's win to strengthen his status as an excellent back-up option to first-choice gloveman Petr Cech.

While Jones, making his first competitive start for Liverpool in almost a year, also made some excellent saves in the Reds' home victory.

Schwarzer, Australia's most capped player, who announced his Socceroos retirement in November, has not made a Premier League appearance since switching to Chelsea from Fulham at the start of the season.

However, he is relied upon by Jose Mourinho in cup competitions and the veteran keeper has not let his manager down.

Monday's clean sheet was his fourth in five starts for the club, with two goals in last month's extra-time League Cup defeat to Sunderland the only blemishes on his record so far as a Chelsea player.

Like Schwarzer, 31-year-old Jones has had to bide his time on the bench as back-up to Liverpool's number one Simon Mignolet.

The Australian's appearance on Monday was his first since a Premier League loss to Southampton in March last year, but he looked sharp, particularly in making one great second-half save to preserve Liverpool's lead.

There was no place in Liverpool's squad for Australian-born teenager Brad Smith, who made his Premier League debut off the bench against Chelsea last month

Oldham's side featured Australian midfielder James Wesolowski, who helped the League One team to a surprise win over Liverpool in the FA Cup fourth round last season.

Elsewhere, rising Socceroos midfielder Tom Rogic was again overlooked as he remained on the bench in Celtic's 4-0 Scottish League win away at St Mirren.